Question 24382: Equal Net Values Do Not Replace Capacity, Conservation or Optimality Duties

Original question and givens

Use vertices=[s6, a6, t6, u6], ordered directed edges=[[s6, a6], [a6, s6], [a6, t6], [t6, a6]], distinct source=s6, sink=t6, aligned capacities=[35, 35, 35, 35] and candidate flows=[28, 14, 21, 7]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).

• G001 — Use vertices=[s6, a6, t6, u6], ordered directed edges=[[s6, a6], [a6, s6], [a6, t6], [t6, a6]], distinct source=s6, sink=t6, aligned capacities=[35, 35, 35, 35] and candidate flows=[28, 14, 21, 7]. Vertices/edges must qualify; capacities are nonnegative. Candidate flows are not assumed feasible. Zip each edge with its own capacity/flow before sorting; never sort those arrays independently. Opposite directions remain distinct; loops and duplicate ordered edges are forbidden. Give (source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-finite-flow-network

Step-by-step derivation

source roles = [flow-value-cross-check]

derived = [[source, sink, edge_capacity_flow_rows, source_net_outflow, sink_net_inflow, net_values_equal, capacity_feasible, conservation_feasible, feasible, maximum_or_min_cut_certificate, boundary]]

source roles = [[s6, a6, t6, u6], [[s6, a6], [a6, s6], [a6, t6], [t6, a6]], [35, 35, 35, 35], [28, 14, 21, 7], s6, t6]

derived = [[a6, s6, t6, u6], [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], 2, Yes]

source roles = [0, [a6, s6], 14, 35]

derived = [Yes, Yes, Yes]

source roles = [1, [a6, t6], 21, 35]

derived = [Yes, Yes, Yes]

source roles = [2, [s6, a6], 28, 35]

derived = [Yes, Yes, Yes]

source roles = [3, [t6, a6], 7, 35]

derived = [Yes, Yes, Yes]

Flow ALL incoming at vertex a6 := sum(28, 7)

Flow ALL incoming at vertex a6 = 35

Flow ALL outgoing at vertex a6 := sum(14, 21)

Flow ALL outgoing at vertex a6 = 35

source roles = [a6, [2, 3], [0, 1]]

derived = [35, 35, Yes, Yes]

Flow ALL incoming at vertex s6 := sum(14)

Flow ALL incoming at vertex s6 = 14

Flow ALL outgoing at vertex s6 := sum(28)

Flow ALL outgoing at vertex s6 = 28

source roles = [s6, [0], [2]]

derived = [14, 28, No, No]

Flow ALL incoming at vertex t6 := sum(21)

Flow ALL incoming at vertex t6 = 21

Flow ALL outgoing at vertex t6 := sum(7)

Flow ALL outgoing at vertex t6 = 7

source roles = [t6, [1], [3]]

derived = [21, 7, No, No]

Flow ALL incoming at vertex u6 := sum()

Flow ALL incoming at vertex u6 = 0

Flow ALL outgoing at vertex u6 := sum()

Flow ALL outgoing at vertex u6 = 0

source roles = [u6, [], []]

derived = [0, 0, Yes, Yes]

Flow source NET outflow not gross outgoing := sub(28, 14)

Flow source NET outflow not gross outgoing = 14

Flow sink NET inflow not gross incoming := sub(21, 7)

Flow sink NET inflow not gross incoming = 14

source roles = [s6, t6, [[[a6, s6], 14, 35, Yes], [[a6, t6], 21, 35, Yes], [[s6, a6], 28, 35, Yes], [[t6, a6], 7, 35, Yes]], [[a6, 35, 35, Yes], [u6, 0, 0, Yes]], 14, 14]

derived = [Yes, Yes, Yes, Yes, No]

Reverse checks

• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute Flow sink NET inflow not gross incoming with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Flow source NET outflow not gross outgoing with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute Flow ALL outgoing at vertex u6 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Flow ALL incoming at vertex u6 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute Flow ALL outgoing at vertex t6 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Flow ALL incoming at vertex t6 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute Flow ALL outgoing at vertex s6 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Flow ALL incoming at vertex s6 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute Flow ALL outgoing at vertex a6 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Flow ALL incoming at vertex a6 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(s6, t6, [[[a6, s6], 35, 14], [[a6, t6], 35, 21], [[s6, a6], 35, 28], [[t6, a6], 35, 7]], 14, 14, Yes, Yes, Yes, Yes, No, feasible_flow_value_cross_check_does_not_claim_maximality_or_min_cut_equality)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.